It is pointed out that explicit Runge-Kutta methods cannot integrate polynomial solutions of degree greater than 1 exactly, except in the case of quadrature problems. Considerations based on this observation lead to the design of a new exponentially-fitted method. The new method is compared with the method of {\it T. E. Simos} [Appl. Math. Lett. 9, No. 6, 61-66 (1996;

Zbl 0864.65052)] and, as for the Simos method, is superior in performance over a classical method for some oscillatory problems. For two further test problems, almost perfect accuracy is achieved by the new method. However, this is evidently because the method integrates problems with purely oscillatory solutions exactly.